real analysis - Lipschitz function and continuously differentiable function

If function $f$ is continuously differentiable at some point, say $x=0$, and is Lipschitz in some neighborhood of $x=0$, is that true there is an open neighborhood of $x=0$ in which $f$ is continuously differentiable?

I know there is a function which is differentiable at just one point and continuous everywhere else. I also know the set of continuity of a derivative of a function is dense. I also familiar with a differentiable function which is not $C^1$ On the Cantor set.